Gordon Brown yesterday warned the terrorist threat to Britain was "long-term and sustained", and said security services and the public will have to remain constantly vigilant.
Mr Brown said it was clear that the attacks in London and Glasgow over the past two days were perpetrated by people associated with al Qaeda.
He urged the public to continue "living their lives as normal" and pledged: "Everything is being done in our powerto protect people's lives".
Mr Brown said that Britain's message to the terrorists must be: "We will not yield, we will not be intimidated and we will not allow anyone to undermine our British way of life."
Mr Brown said the terror attacks were being carried out by "a group of people not just in this country but around the world who are prepared to inflict maximum damage on civilians" in pursuit of their aims.
The response to them must be not only on a military, police and security level, but must also include measures to win the "hearts of minds" of ordinary Muslims and separate the extremists from the mainstream majority.
He played down suggestions that the terrorists were motivated by anger over British involvement in Iraq and Afghanistan, saying that instead they seemed to have a "grievance against society, particularly against the values that we represent and the values decent people of all religions represent".
He told Sunday AM: "Irrespective of Iraq, irrespective of Afghanistan, irrespective of what is happening in different parts of the world, we have an international organisation trying to inflict the maximum damage on civilian life in pursuit of a terrorist cause that is totally unacceptable to most people."
The Prime Minister gave no indication that the authorities had received intelligence of an attack being planned to mark his arrival at 10 Downing Street. He said: "We have got to recognise what the nature of the threat that we are dealing with is.
"Al Qaeda and people who are related to al Qaeda, and while I don't want to comment on the police investigation that is on-going, it is clear that we are dealing in general terms with people who are associated with al Qaeda.
"It wants to make its point, its propaganda effort, by inflicting maximum damage, irrespective of religion, on civilian life.
"In any country, and we know al Qaeda are operating in more than 60 countries, we can expect that they will use different forms of missile or weapon, or different forms of activity, whether it is planes or cars, to inflict that damage."
Mr Brown dismissed claims that the terrorists could find justification for their crimes in the Islamic religion.
"Terrorism can never be justified as an act of faith," he said. "It is an act of evil in all circumstances."
Jacqui Smith, the new Home Secretary, is due to make a statement on the government's response to terrorist attacks in the House of Commons this afternoon.
Ms Smith, who hardly had her feet under the desk when the terrorists struck in London in the early hours of Friday morning, emerged from a government meeting yesterday to appeal to the public to remain vigilant.
While she revealed that there had been some important developments overnight, she would not release details, declaring it was up to the police to update the public when they thought it necessary.
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